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\title{\vglue -45pt  \LARGE \bf   About Snail Shell Surfaces  \vglue -15pt }
\author{ Traudel Karcher}
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\LARGE



These snail-like surfaces are included for their entertaining shapes. 
Try making one of your own.

In spite of their complicated appearance, the snail surfaces are constructed
as one-parameter families of circles $u \mapsto C_v(u)$. First we introduce two
auxiliary variables. The surface parameter $v$ is changed by a quadratic
term that permits closing the snails at the top. The parameter ee controls
the size of the opening of the snail (default $ee=-2$):

$ v := v + (v + ee)^2/16 $.

The second variable controls the radius of the circles:

$ s := \exp(-cc\ vv).\  $ (Note that $s$ is a function of $v$.)

The circles $u \mapsto C_v(u)$ of radius $bb\cdot s$ lie at first in an $r$-$y$-plane:

$r := \ s\ aa + s\ bb\  \cos(u)),$

$y := dd(1-s) + s\ bb\  \sin(u).$

The parameter $dd$ controls the length of the snail from top to bottom. And
the remaining coordinates in $\Bbb R^3$ are given by

$	x := r \cos(vv) ,$

$	z := r \sin(vv) ,$

so that the plane of the circle $C_v$ also rotates with $v$.

Make only {\bf small} changes to $cc$ and keep $bb \ge aa$.

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